Question: $ A = \left[\begin{array}{rrr}4 & 3 & 0 \\ 4 & 2 & 0 \\ 0 & 1 & 0\end{array}\right]$ $ E = \left[\begin{array}{rr}-2 & -2 \\ -2 & -2\end{array}\right]$ Is $ A E$ defined?
Solution: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ A$ , have? How many rows does the second matrix, $ E$ , have? Since $ A$ has a different number of columns (3) than $ E$ has rows (2), $ A E$ is not defined.